Real Congruence of Complex Matrix Pencils and Complex Projections of
Real Veronese Varieties

article abstract

Quadratically parametrized maps from a real projective space to a
complex projective space are constructed as projections of the
Veronese embedding.  A classification theorem relates equivalence
classes of projections to real congruence classes of complex symmetric
matrix pencils.  The images of some low-dimensional cases include
certain quartic curves in the Riemann sphere, models of the real
projective plane in complex projective 4-space, and some normal form
varieties for real submanifolds of complex space with CR
singularities.